Applications of Laplace’s Method in Asymptotic Analysis
Keywords:
Laplace’s method, large-parameter functions, asymptotic analysis, asymptotic formula, Stirling–Legendre formula, Wallis formula, integral estimatesAbstract
This article investigates the application of Laplace’s method in asymptotic analysis. An asymptotic formula is derived showing that, as the parameter increases, the value of the integral involving powers of a function is mainly determined by the behavior of the function near its maximum point. The paper also discusses the asymptotic properties of large-parameter functions and presents related analytical results.
References
G. Pólya and G. Szegő, Problems and Theorems in Analysis, vol. 1. Moscow, Russia: Nauka, 1978.
P. Laplace, Philosophical Essay on Probabilities. Moscow, Russia: LIBROKOM, 2011.
G. M. Fikhtengolts, Course of Differential and Integral Calculus, vols. 1–3. Moscow, Russia: Nauka, 1970.
R. Engelking, General Topology. Moscow, Russia: Mir Publishers, 1986.
I. Juhász, Cardinal Functions in Topology – Ten Years Later. Amsterdam, Netherlands: Mathematical Centre, 1980.
A. A. Borovkov, Theory of Probability. Moscow, Russia: Nauka, 1982.
A. Abdushukurov and T. Zuparov, Probability Theory and Mathematical Statistics. Tashkent, Uzbekistan: Bo‘ston, 2015.
M. Ya. Kelbert and Yu. M. Sukhov, Probability and Statistics in Examples and Problems, vol. 2. Moscow, Russia: MCCME, 2007.
S. B. Iskandarov, “Applications of the Method of Large-Parameter Functions in Asymptotic Analysis. Laplace’s Method,” Ilm Sarchashmalari, no. 11, pp. 3–8, 2023.
R. Wong, Asymptotic Approximations of Integrals. Philadelphia, PA, USA: SIAM, 2001.
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals. New York, NY, USA: Dover Publications, 1986.
F. W. J. Olver, Asymptotics and Special Functions. New York, NY, USA: Academic Press, 1974.
A. Erdélyi, Asymptotic Expansions. New York, NY, USA: Dover Publications, 1956.
M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations. Berlin, Germany: Springer, 1993.
N. G. de Bruijn, Asymptotic Methods in Analysis. Amsterdam, Netherlands: North-Holland Publishing Company, 1981.
E. J. Hinch, Perturbation Methods. Cambridge, U.K.: Cambridge University Press, 1991.
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. New York, NY, USA: McGraw-Hill, 1978.
